The explanation is somewhat simple. Regulation (EU) 2015/751 of 29 April 2015, which capped the interchange fees for four-party schemes like Visa and Mastercard, was according to a World Bank presentation, specifically designed to
Put an end to the race between MasterCard and Visa for higher interchange fees (reverse
competition)
and
Introduce more competition
on the acquiring side
Why three-party schemes (like Amex) were excluded is a bit harder to fathom. I suspect some lobbying was at play. There was another (temporary) exception allowed:
Three party schemes with licensees: Member States can
exempt for till 09.12.2018 if whole scheme has <3% market
share
Another presentation has this interesting slide, which shows that in the run-up to the Regulation, Visa and Mastercard (but not Amex) were sued in the EU (UK at least) by retailers for the fees. So I guess the EU took the minimal measures with the 2015 regulation to avoid the exact same lawsuits, but not similar ones (which predictably enough followed).

A bit more searching found a US-EU comparative paper on card regulations that states that Amex (with rewards) is mostly used by rich people... so I guess that the EU didn't see a need to help them (with the fees). Also, and perhaps more importantly, it seems that the EU considers that steering the consumer to cards with lower fees is feasible and even desirable in the case of three-party schemes, unlike in the case of four-party schemes in which the Visa/Mastercard as a sort of intermediary monopoly. (Frankly, this is seem to work in some EU theory, in which there could be many three-party scheme, even though in practice there probably aren't anymore than four-party schemes, i.e. the three-party market is just as oligopolic.)
The inspiration for that rule in the PSD1 Directive also came from competition policy thinking: "[i]n order to promote transparency and competition", "[t]he payment service provider shall not prevent the [store] from requesting from the payer a charge or from offering him a reduction for the use of a given payment instrument".
Two conclusions can be drawn from this. First, prohibiting corporate behaviour by regulation is arguably even more severe than characterising it as a "restriction by object" under Article 101. A restriction by object may be exempted through the efficiency defence under Article 101(3), while EU regulation of anti-steering provisions does not allow for exemptions.
Second, in the eyes of the European Commission and EU legislator, allowing "steering" exposes "more expensive means of payment" to more competition, while anti-steering policies cause all consumers to subsidise the wealthier users of high-cost cards. This rationale for prohibiting anti-steering clauses in the EU would apply equally to the Amex case.
Since Amex cards function on the basis of high rewards but also high fees to cardholders, Amex cardholders tend to be wealthier than other cardholders. Amex cards also charge higher fees to stores than Visa and Mastercard. Stores internalise those fees on Amex transactions as costs ultimately borne by all customers. Therefore Amex cardholder rewards are subsidised by all customers. In technical terms, one could call this a negative externality. In simple terms, one would call this a mechanism of economic inequality – a "reverse Robin Hood payment system".
The same idea, more concisely stated in an industry paper (from the merchants' perspective):
There are a number of exemptions from the caps:
- Commercial cards: the definition is intended to limit this exemption to cards which are for true business use only;
- 3-party schemes (i.e. Amex, Diners) unless they operates under a license or through an agent, in which case it is treated as a 4-party scheme and caps apply. There is also a member state option, which enables full exemption of 3-party schemes to a market share of 3% for a limited period.
The rationale here is that these exemptions, coupled with the steering provisions in Article 11 and the near removal of the honour all cards rule, will enable merchants to steer consumers to the most efficient payment type.
Finally, Amex has very low market share in the EU (1.3% in 2014), so the regulators probably didn't see a huge concern with it unlike for Visa and Mastercard, which each had some ~43% market share in the EU that year (42.9% and 42.6% to be more precise). These are stats by number of payments. The figures are somewhat different by value of transactions, but still low for Amex (2.1%; Visa 53.7%, Mastercard 38.7%).
Interestingly, an industry article notes that Amex found itself at a severe disadvantage following the new rules, and has subsequently largely exited the EU market in 2019, with a few [sectoral] exceptions. This was actually fairly predictable following their loss of a 2018 lawsuit,
The popular British Airways American Express (Amex) credit card is among those which could disappear from the UK market after the European Court of Justice (ECJ) this week issued a ruling against the card provider. [...]
But this week’s European-wide ruling has judged that in cases when a card is co-branded, such as with an airline, this counts as a fourth party and thus fees charged to retailers must be capped.
And as a result:
Over the course of 2019, American Express credit cards issued under independent operator agreements will be rendered invalid in all countries of the European Union. Various banks that have up to now had exclusive licensing contracts with American Express have already responded accordingly and provided their clients with the opportunity to switch to other card brands.
The card company is still open to dealing directly with interested individuals and has also opted to maintain a physical presence in several European Union countries, operating its proprietary businesses or joint ventures, including France, one of its most lucrative markets. [...]
So it looks like the three-party exception (to the fee cap) is something that Amex actually likes, as they chose to withdraw from any card business where the four-party rules were judged to apply (but insofar not from the three-party ones).
Also interestingly, Mastercard argued that the same rules should apply to three-party schemes (but they didn't get that from the EU insofar):
As demonstrated by the experience in Australia, if the implicit interchange in pure three party
schemes was not regulated to the same extent, this would likely result in an increase in the use of
unregulated three party proprietary card schemes. Given that the three party schemes are generally
more expensive to the merchant and the cardholder than four party card schemes, this would run
directly contrary to the EC’s stated objective.
What's also interesting from another Mastercard 2014 lobbying document, is that EU apparently considered regulating the three-party schemes likewise at some point, but that apparently changed during the regulation process:
We would support amendments to the proposed Regulation that:
Maintain the European Commission’s original proposal to consider three party schemes using payment service providers as
issuers or acquirers as four party payment schemes and follow the same rules